And although the government-to-government relationship is between the Federal government and each of the individual tribal nations, not any collective group of those nations, the sheer number of the sovereign tribal entities probably demands some sort of alliance or collective entity representing the tribes. This is important for avoiding conflicting requests for policy changes, and for keeping some semblance of unity in the causes.
This issue was taken up in 1993, when NCAI leaders urged the Congress and Administration to authorize the study of arrangements that would improve tribal consultation with both entities. The NCAI had proposed formation of a Inter-Departmental Council on Indian Affairs (IDCIA), to consult with the Administration; and a National Native American Advisory Commission (NNAAC), to advise Congress.
But one council or committee for the executive branch and one for the legislative branch doesn't seem like enough. How about a series of regional committees for regional issues such as land or water rights? And a series of subject-oriented committees for tribes with subject-oriented issues: criminal justice, healthcare, gaming, etc.? These committees would have the power to negotiate on behalf of the tribes. Once they negotiated a policy, the committee would take it back to the tribes for a vote.
I'm sure there are a slew of negotiating bodies already. The difference is empowering them to negotiate officially. Perhaps with an arbiter such as a special court, with Native and non-Native judges, to rule on any disputes.
For more on the subject, see Obama's Memo to His Cabinent and Obama at the Tribal Summit.